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batconjurer
batconjurer
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Suppose you have a unitary matrix $U$ such that $\overline{U}U=D$ for some diagonal unitary matrix $D$ (everything is taking place over $\mathbb{C}$). This is equivalent to a couple of other conditions:

  1. $DU=U^T$

  2. $DUD=U$

Furthermore, this implies that $U^2$ is symmetric. So most unitary matrices do not satisfy this property. But when they do, is it true that $D^2=I$, which is equivalent to saying that $D$ commutes with $U$.?

Suppose you have a unitary matrix $U$ such that $\overline{U}U=D$ for some diagonal unitary matrix $D$ (everything is taking place over $\mathbb{C}$). This is equivalent to a couple of other conditions:

  1. $DU=U^T$

  2. $DUD=U$

Furthermore, this implies that $U^2$ is symmetric. So most unitary matrices do not satisfy this property. But when they do, is it true that $D^2=I$, which is equivalent to saying that $D$ commutes with $U$.

Suppose you have a unitary matrix $U$ such that $\overline{U}U=D$ for some diagonal unitary matrix $D$ (everything is taking place over $\mathbb{C}$). This is equivalent to a couple of other conditions:

  1. $DU=U^T$

  2. $DUD=U$

Furthermore, this implies that $U^2$ is symmetric. So most unitary matrices do not satisfy this property. But when they do, is it true that $D^2=I$, which is equivalent to saying that $D$ commutes with $U$?

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batconjurer
batconjurer
  • 928
  • 4
  • 11

When a Unitary matrix times its complex conjugate is diagonal

Suppose you have a unitary matrix $U$ such that $\overline{U}U=D$ for some diagonal unitary matrix $D$ (everything is taking place over $\mathbb{C}$). This is equivalent to a couple of other conditions:

  1. $DU=U^T$

  2. $DUD=U$

Furthermore, this implies that $U^2$ is symmetric. So most unitary matrices do not satisfy this property. But when they do, is it true that $D^2=I$, which is equivalent to saying that $D$ commutes with $U$.